The overall objective of this study is to develop new stochastic micromechanical and multiscale models for elastic properties and fracture analysis of functionally graded materials (FGMs) involving random microstructure and constituent material properties. To meet the objective, five major research directions have been defined: (1) development of a new stochastic micromechanical model for predicting probabilistic characteristics of effective elastic properties; (2) development of new stochastic multiscale models for two-dimensional fracture analysis; (3) performance of a parametric study to evaluate the effects of microstructural parameters on crack-driving forces; (4) development of a new polynomial dimensional decomposition method to account for discrete particle locations in stochastic fracture, and (5) extension of the stochastic concurrent multiscale model for solving three-dimensional fracture problems.;First, a new stochastic micromechanical model was developed for predicting probabilistic characteristics of elastic mechanical properties of an isotropic FGM subject to statistical uncertainties in constituent material properties and their respective volume fractions. The model provides both accurate and computationally efficient estimates of probabilistic characteristics of effective FGM properties. Second, three multiscale models, comprising sequential, invasive, and concurrent models, were developed for fracture analysis of a two-phase FGM. The models involve stochastic descriptions of microstructural features and constituent material properties; a two-scale algorithm including microscale and macroscale analyses for determining crack-driving forces; and two stochastic methods for uncertainty propagation. Results indicate that the concurrent multiscale model is sufficiently accurate, gives probabilistic solutions very close to those generated from the microscale model, and can reduce the computational effort of the latter model by more than a factor of two. Third, employing the concurrent multiscale model, a parametric study on fracture behavior of FGMs was performed. The study involves stochastic descriptions of FGM microstructure and constituent elastic properties and limited crack-propagation simulations. Fourth, a computationally efficient polynomial dimensional decomposition method was developed to accurately capture the fracture results obtained using the concurrent multi-scale model. Finally, the newly developed efficient stochastic method was employed to solve a three-dimensional fracture problem. Results indicate that the crack-driving forces can vary significantly along the crack front.
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